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Chashokreddy18@gmail.com posted an Question
July 29, 2020 • 16:05 pm 30 points
  • CSIR NET
  • Mathematical Sciences

Gt - can you please explain the following problem? (q 30 , p 15)

2 Answer(s) Answer Now
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  • Shashi ranjan sinha

    I just have a single request... please always take a look on my answers and in case of doubts, I ll try my best to clarify it

  • Chashokreddy18@gmail.com Tiwari

    Thank you sir for explaining, I understood.

  • Shashi ranjan sinha Best Answer

    S3 is a permutation group on say 3 symbols 1,2,3. Then (123) represents a cyclic permutation of 3 length and so is of order 3.....In Z2, 1 bar denotes the residue class of 1, that is, it is an equivalence class consisting of those integers which when divided by 2 lefts 1 as the remainder

  • Shashi ranjan sinha

    1) Order of S3 is 6 and that of Z2 is 2, therefore S3 X Z2 is of order 12...... 2) S3 is non abelian implies S3 X Z2 is non abelian....3) Any non abelian group of order 12 is either isomorphic to A4 or D6.... 4) Any two isomorphic groups has same number of same ordered elements. Since A4 has no element of order 6, while S3 X Z2 has , therefore they can't be isomorphic. ..... Hence, S3 X Z2 is isomorphic to D6

  • Shashi ranjan sinha best-answer

    see the attachments.

    cropped1342360520793045066.jpg
    cropped958276853549142397.jpg
    eduncle-logo-app

    When we divide any integer by 2, it leaves either 0 or 1 as the remainder....set of all the Integers which leaves zero as the remainder is denoted by 0 bar and set of those integers which leaves 1 as the remainder is denoted by 1 bar.... Z2 is two element set which forms a group with respect to addition of residue classes

    eduncle-logo-app

    However some books avoid this notion and they don't put bar over 0 and 1.... but to give u sense of the fact that 0 and 1 in Z2 are not integers but are specific sets, I put bar over it

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